This paper describes a generalization of the virtual redundant axis (VRA) method previously introduced by the authors for robust, position-controlled inverse kinematics of serial robots passing through or close to a singular configuration. While the VRA method regards only one body- or inertially-fixed virtual axis, the new approach can handle multiple “floating” virtual axes which are smooth functions of the robot configuration and are not necessarily fixed to the robot links or base. This allows for robustly negotiating inverse kinematics even in the presence of multiple singularities with a single parameter set, while concentrating tracking errors solely in the locked directions. The paper discusses the theory, including a novel Jacobian form with non body-fixed joint axes, validating its performance on a real industrial robot Kuka KR15/2 and comparing it to the weighted damped least squares method (WDLS).